This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic α with 0 < |α| < 1, the field ℚ(F(α),F(α⁴),G(α),G(α⁴)) has transcendence degree 3 over ℚ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-2,
author = {Peter Bundschuh and Keijo V\"a\"an\"anen},
title = {Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {239-249},
zbl = {06414109},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-2}
}
Peter Bundschuh; Keijo Väänänen. Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II. Acta Arithmetica, Tome 168 (2015) pp. 239-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa167-3-2/